4.1 Sums and Differences

Linear Operation Rules

  • There are several rules in differentiation. Let’s start with these:

i) If f(x)=c where c is a constant, then f^{\prime}(x)=0 (Constant function)

ii) If f(x)=k \cdot g(x), then f^{\prime}(x)=k \cdot g^{\prime}(x) (Multiple)

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1.4 Algebra of Limits (Not Required)

  • Any materials in this notes is not required in the scope of the course, but is included for your understanding.

Algebra of Limits

  • Below are some properties of limits, assuming both \lim _{x \rightarrow a}f(x) and \lim _{x \rightarrow a}g(x) exists:

i) Sum:

\lim _{x \rightarrow a}[f(x)+g(x)]=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a}g(x)

ii) Difference:

\lim _{x \rightarrow a}[f(x)-g(x)]=\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)

iii) Multiple:

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5.5 Sum and Product of Functions

Sum and Product of Functions

  • Different functions can be added or multiplied together.
  • Say there is two functions f(x) and g(x). We write their sum as (f+g)(x), and product as (fg)(x). In particular, f+g is the sum, fg is the product.
  • Thus, to be clearer, it simply means

\begin{aligned} (f+g)(x) &=f(x)+g(x) \\(f g)(x) &=f(x) g(x)\end{aligned}

  • When performing such functions, we have to be aware of the domain. For the sum and product to be defined, the domain of this combined function must be in both the domain of f and g.
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